I.
Federated Square Root Filter for
Decentralized Para1le1 Processes
NEAL A. CARLSON
Integrity Systems, Inc.
An efficient, federated Kalman filter is developed for use in
distributed multisensor systems. This new design accommodates
sensor-dedicated local filters, some of which use data from a
common reference subsystem The local filters run in parallel,
and also provide sensor data compression via prefiltering. The
master filter runs at a selectable reduced rate, fusing local
filter outputs via efficient square root algorithms. C o n "
local
process w
i
s
e correlations are handled by use of a conservative
matrix upper bound. The federated filter yields estimates that are
globally optimal, or conservatively suboptimal, depending upon
the master filter processing rate. This design achieves a major
improvement in throughput (speed), is well suited to real-time
system Inplementation, and enhances fault detection, isolation
and recovery capability.
Manuscript received April 2,1989.
IEEE Log No. 33466.
This work was supported by the Defense Small Business Innovation
Research (SBIR) Program under Contract F33fj15-86-C-1087,
administered by the Avionics Laboratory, WRDC/AAAN-2,
Wright-Patterson Air Force Base,Dayton, Ohio.
This work is based on a paper of the same title presented at
NAECON '87, May 18-22, 1987, and printed in the Record of that
conference.
Author's address: Integrity Systems, Inc., 600 Main St., Suite 4,
Winchester, MA 01890.
0018-9251/90/0500-0517 $1.00 @ 1990 IEEE
INTRODUCTION
This paper develops a federated Kalman filter
architecture applicable to decentralized sensor
systems with parallel processing capabilities. This
architecture provides significant advantages for
real-time multisensor applications such as integrated
navigation systems. While multisensor systems embody
the potential for high levels of accuracy and fault
tolerance, that potential has not been fully realized
via past application of classical Kalman filtering
techniques.
Classical techniques applied to multisensor systems
can yield severe computation loads when implemented
in strictly optimal fashion. Conversely, ad hoc
simplifications are not always reliable, being subject to
poor accuracy, instability, and even divergence under
certain operating conditions.
For these and other reasons, there has been
considerable recent interest in the development
of decentralized (or distributed) Kalman filter
architectures. The advent of parallel processing
technology and emphasis on fault-tolerant system
design are additional factors motivating such
development. While several different approaches to
decentralized filtering have been developed, none
appear to have been implemented in real-time system
applications, e.g., aircraft navigation systems.
Theoretical approaches to decentralized filtering
were developed by Speyer [l], Chang [2], Willsky et al.
[3], Levy et al. [4],and Castanon et al. [5].However,
these early approaches are generally not suitable or
practical for real-time estimation of time-varying
systems, due to restrictive system assumptions or large
data transfer requirements.
A more recent and flexible decentralized filtering
method was developed and implemented by Bierman
[6], for filtering and smoothing of satellite orbit data.
This method constructs a set of independent (local)
state estimates, which can be optimally combined
in straightforward fashion. The local state estimates
are naturally disjoint with respect to their local
measurement sets. They are then constructed to be
disjoint with respect to common initial conditions
and process noise, by assigning 100 percent of the
associated information to one local estimate, and
zero (infinite covariance) to the others. Thus no
cross-correlations exist, and the local estimates can
be optimally combined by a master filter via simple
addition of the local information.
While this method is both optimal and efficient
on a per-cycle basis, it does require the master filter
to operate at the maximum local measurement rate.
The local filters with infinite process noise have no
memory, losing all process information between
measurement cycles. Hence they do not perform
recursive filtering, but yield single-point least-squares
solutions. This aspect of the method is of no concern
IEEE 'IRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 26, NO. 3 MAY 1959
517
in its intended application, but is undesirable in those
applications that require either a) physically meaningful
local filters with individually usable results (e.g.,
for local sensor rate-aiding) or b) master filter rate
reduction via local sensor data compression.
A conceptual decentralized filtering structure
was proposed by Kerr [7],in which several sma!ler
filters run in parallel and process data from separate
navigation subsystems (e.g., Global Positioning System
(GPS)). The outputs of these local subsystem filters
are to be combined by a master “collating” filter;
however, no mathematical basis for that filter was
presented. Given a sound basis, this decentralized
filtering architecture would provide distinct advantages,
e.g., asynchronous operation, fault detection and
isolation, and reconfigurability.
Recently, Hashemipour et al. [SI developed new
parallel Kalman filtering structures for multisensor
networks amenable to parallel processing. While this
method yields globally optimal estimates and a linear
speed-up rate, it requires significant data exchange
among the filter components, and does not use local
data compression to reduce the master rate.
The new federated filter design developed here
achieves the inherent advantages of distributed systems
by means of a relatively simple yet pivotal extension of
Bierman’s method [6]. This extension provides usable
and physically meaningful local filters, and allows
master filter rate reduction via local data compression
(prefiltering). It yields globally optimal or suboptimal
estimation accuracy, as a function of the selectable
master filter rate, with a high degree of fault tolerance.
The remaining sections of this paper describe
the distributed filtering problem (Section 11), the
new federated filter structure (Section 111), its
covariance square root form (Section IV), alternative
implementations (Section V), current and future
applications (Section VI), summary and conclusions
(Section VII), and formal mathematics (Appendix).
II.
PROBLEM STATEMENT
In concept, federated filtering is a two-stage data
processing technique in which the outputs of local,
sensor-related filters are subsequently processed and
combined by a larger master filter, as illustrated in Fig.
1. This figure shows the major flow of information,
but does not attempt to represent all the possible
data exchanges among components. As suggested by
this figure, each local filter is dedicated to a separate
sensor subsystem. One or more local filters may also
use data from a common reference system, e.g., an
inertial navigation system.
This general structure applies to two federated
system types of interest, denoted “A’ and “B”. Type A
systems involve fixed local filter designs that have been
developed elsewhere for stand-alone operation, without
regard for federated applications. While their basic
518
d.
f
’
R
REFERfMCE
SYSTEM
d,
!
YASEA
FILER
LOCAL
SENSOR I
1111111 1
LOCAL
LOUL
SENSOR 2
I
p--Lq.-..kq
SENSOR Y
FlLlER N
Fig. 1. Federated filter structure.
designs are fixed, some internal model parameters can
be modified via initial data loads. Type A systems are
of interest because of the need to use existing local
filter designs (e.g., aided GPS) over the next few years.
Type B systems involve totally flexible local
filter designs that can be tailored to best support
federated filtering operations. Type B systems permit
all the federated filter components to be designed
for cooperative operation, with design decisions
based on global considerations rather than on local
considerations alone. Type B systems are of interest
because they represent the idealized case toward which
future federated designs may evolve.
The federated filter structure can be developed
in terms of an optimal, linear estimation problem as
follows. First, consider a system state vector x that
propagates from time point t’ to t according to the
following dynamic model:
x = ax’
+ Gu.
(1)
Here, @ is the state transition matrix between time
points t’ and t , G is the process noise distribution
matrix, and U is the additive uncertainty vector due
to white process noise acting over the timestep. The
initial state estimate So and the sequential values u ( t k )
are uncorrelated, per the following error statistics:
SO = xo
+ eo;
E [ d ]= 0
E[uke“] = 0
~ [ e ’=
] 0;
~ [ e ’ e ’=
~ ]PO
E[ukuJT]=
+
(2)
(3)
E[ukejT]= 0,
k
>j.
(4)
Our system also has access to external
measurements 5 from i = 1, N separate local sensor
subsystems. Measurements from different local sensors
are independent, and comprise disjoint data sets. The
discrete measurements from sensor i at time t are
linearly related to the true state x:
2; = H j X
+
vj
(5)
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 26, NO. 3 MAY 1990
(21 - x1)
uncorrelated, per the following error statistics:
Q=
E[$] = 0;
E [ v ~=
v @bkJ
~]
(6)
(13)
(gN-xN)
p-1
Ill. FEDERATED FILTER STRUCTURE
This section develops the theoretical basis for the
new federated filter structure. Section IV then provides
an efficient square root implementation of this new
structure.
We first define a composite global filter, then
partition its operations into independent local subsets.
Consider the following composite state vector and
corresponding covariance matrix definitions:
1x1 1
local partition 1
X =
(9)
local partition N
common system states
(i version)
sensor i bias states
In this case, the globally optimal solution is given by
the following familiar result (where, for simplicity, we
omit the effects of the noncommon states):
This simple, optimal result is of real interest. Our
new federated filter will be constructed to yield local
estimates that are in fact uncorrelated; the above
equations then represent the optimal combination of
those local filter solutions.
Consider next the globally optimal processing of a
local measurement from sensor i, written in terms of
the composite global variables:
(10)
2; = H x + v ; ;
H = [0...Hi ...01
P i = Pjk - P;iH,'A-'H;P,?;:
(12)
The full global state vector (9) is the composite of
the N local state partitions xi. As indicated by (lo),
each local partition contains the common system state
elements, plus unique bias states for its own sensor.
The composite global state contains N versions of
the common system state x,. This redundancy causes
no theoretical difficulty, and is optimally resolved
below. The composite covariance matrix P can contain
cross-partitions Pi; as well as the local partitions Pii.
Now, given a set of N local state estimates f; and
their composite covariance P , the globally optimal
estimate off the full system state x minimizes this
(17)
(20)
Equations (17) to (20) with j = i indicate that sensor
i measurements affect the i state and covariance
as if only i existed. Furthermore, i measurements
do not affect other local states j # i unless the
cross-covariances P;i are nonzero. Note that, if
P;; is initially zero, then it stays zero, and other
local partitions P,; plus all Pjk cross-covariances
are unchanged. Thus, if the local filter states are
initially uncorrelated, then the local measurement
sets can be processed independently, and they remain
uncorrelated.
Consider next the global time propagation step.
This step comprises the crux of the new federated
filter method. The full global state and covariance
CARLSON: FEDERATED SQUARE ROOT FILTER FOR DECENTRALIZED PARALLEL PROCESSES
519
propagation from t’ to t can be described as follows:
+
[Cl]
U
LGN
@Tl
..
(21)
J
The upper bound in (25)is “more positive definite”
than the original matrix. That is, the upper bound
minus the original matrix (the difference matrix)
is positive semidefinite. Substituting this result in
(22) then yields the following upper bound on the
propagated global covariance P :
”’
..
0
In (21) and (22), the global state transition matrix
@ is block diagonal because each local filter state is
dynamically self-contained (the redundant common
states do not affect one another). However, the
common system states (redundant local estimates)
are driven by the same process noise, as indicated.
The common process noise U serves to cross-correlate
the separate local filter estimates, even if they were
originally uncorrelated:
pji = + j j ~ ; i + ;
+ G~QGT
(23)
We next make some modifications to the
global representation of the common process noise
covariance. First, we rewrite the process noise term
in (25) as
LGN
J
Thus, setting the global process noise and
propagated state error covariances to their upper
bounds above, we obtain the following partition
results:
pi;= @ii~,!i@; +
G~~QGT
p . . = @..P!.@..
I’
-0
if
Pji = 0.
(29)
(30)
Equations (29) and (30) represent a conservative
result for the global time propagation step, in that the
process noise covariance is replaced by a larger, upper
bound. Thus, each local filter relies somewhat less on
the propagated state value, and somewhat more on the
latest measurements (consistent with good filter design
practice, if not overdone).
A similar upper bound to that of (25) can be
placed on the initial value of the state covariance
matrix, which is the other common element across the
local partitions. This bound likewise yields disjoint,
conservatively larger values for the local filter initial
covariances, also multiplied by the factor 7; (normally
NI.
The net result of this upper-bounding approach for
the state and process noise covariance matrices is quite
significant:
(24)
Now, the N x N Q-matrix on the right side has the
following upper bound (provable via determinants; also
see Appendix A):
520
1) initial covariance matrix bounds are uncorrelated by
construction;
2) process noise covariance bounds are uncorrelated
by construction;
3) the global state transition matrix introduces no
cross-correlations;
4) local measurement updates introduce no
cross-correlations;
5) independent local filter estimates can be combined
to yield a globally optimal solution via the relatively
simple method of (15) and (16).
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 26, NO. 3 MAY 19W
Thus we have devised a conservative procedure
whereby the global filter can be partitioned into a
set of independent local filters, whose outputs are
periodically combined by a master fusion filter.
While the above approach to bounding the
common process statistics may seem rather heuristic,
it can be rigorously justified (see the Appendix).
Furthermore, it yields the globally optimal estimate
when applied via the following procedure, where the
average y; factor is simply N:
1) initial local covariances are set to 7; x the common
system value;
2) local filters use 7; x the common process noise
covariance value;
3) local filters process own-sensor measurements via
locally optimal (Kalman) algorithm;
4) master filter combines local filter solutions after
each update cycle per (15) and (16);
5) master filter resets local filter states to master
value, and local covariances to 7; x master value.
For example, consider three local filters that share
the same initial state estimate and the same process
noise. Each local filter multiplies the common initial
covariance and process noise covariances by y; = 3
(dividing the information by 3). After several time
steps, the three local filter solutions are combined
as “independent” estimates via (15) and (16). Their
information is thus summed, yielding 3 x 1/3 = 1.0
times the correct value.
Next, each local filter incorporates measurements
from its own sensor. Since these data sets are naturally
disjoint, the correct information sum again results from
their combined solutions. However, if the local filters
pass through several time/measurement cycles before
the master filter combines their solutions, then some
of the available information is lost, and the master
filter estimate will be conservatively suboptimal (see
the Appendix).
In summary, the new federated filter design and
operating procedures to obtain globally optimal
estimation performance are these:
1) design local filters for stand-alone operation using
local sensor measurements and reference system
data (if locally needed);
2) multiply local filter covariances by y; for common
initial estimates and process noises (usually y; = N,
the number sharing common information);
3) combine local filter state estimates via optimal
fusion algorithm;
4) reset local filters with fused state and y; times the
fused covariance.
IV.
the federated filter structure in square root form. We
can choose either covariance square root form, or
information square root form, or a mix, to suit any
particular application.
Covariance form is recommended here for
real-time system applications. It is computationally
advantageous when actual state and covariance
outputs are required frequently, e.g., every filter cycle.
It also readily supports fault detection via tests of
measurement residuals (actual minus predicted values)
as a normal part of the update process. However,
other applications such as satellite orbit determination
may favor information form.
We also recommend the P = UDUT factorization
[9] for mechanizing square root filter algorithms.
However, the equivalent square root representation
P = SST [lo]is much simpler to use for analytical
development. Once results have been derived in terms
of S, they can readily be implemented in terms of the
U-D factors.
We first define matrix square roots of the
state error covariance P ( n x n), the process noise
covariance Q ( p x p), and the measurement noise
covariance R(m x m):
S = p1/2;
V = R’12.
W = Q1/2;
The following equations present the recommended
mechanization of the new federated filter structure in
covariance square root form, first the local filters and
then the master filter. The symbol TU represents an
orthogonal transformation operator, constructed to
reduce the preceding matrix to upper triangular form
and/or the minimum number of columns, via successive
row rotations [9].
Local Filters (i = 1,N )
(from t’ to t )
9; = @ i ; i :
(32)
(33)
(34)
a; = a; + J;C;’(z; - f i g ; ) .
(35)
Master Fusion Filter (m)
2, = amma;
[Sm
01 = [ @ m m S L
Start: 2, = XI;
Do for i = 2 to N:
H, =
(from t’ to t )
GmWmlT~r
S , = SI.
[? ; ;I;
I;i
(36)
(37)
(38)
v, =s;
FEDERATED SQUARE ROOT MECHANIZATION
To maximize computational efficiency, numerical
stability, and effective precision, we wish to implement
(31)
(39)
(40)
2; = 9,
+ J m C i l ( a ;- H,?,).
CARLSON: FEDERATED SQUARE ROOT FILTER FOR DECENTRALIZED PARALLEL PROCESSES
(41)
521
Finish: for i = 1, N :
(43)
In (33), the multiplier y;l2 applies only to
the common part of Wj. In (39), etc., LCi is a
transformation matrix that allows for possible
coordinate frame differences among the filters. In (43),
subscripts c and b; refer to the common and sensor-i
bias states. The scalar multipliers yfI2 will normally
equal “I2, but can differ per (26). Note that the
propagated master filter solution (36, 37) is not used
in the optimal fusion algorithm (38-41); however, it
has other uses. Si is lower triangular in (43).
V.
IMPLEMENTATION ALTERNATIVES
The implementation of the new federated filter
presented in the previous section yields the globally
optimal solution to the distributed estimation problem.
However, there are many practical applications in
which global optimality (maximum estimation accuracy)
is not required. In such cases, the system designer
may prefer to give up some accuracy in order to gain
throughput (computation speed), fault tolerance, or
real-time system simplicity.
Several alternative implementations of the
new federated filter architecture can be devised to
accommodate system design criteria besides strict
optimality. In general, the new architecture provides
a useful framework for assessing the tradeoffs among
competing design criteria. The following paragraphs
address some of them.
Data CompresswnlRate Reduction: The new
federated filter architecture in its fully optimal form
already provides an N-fold speed-up over a single,
global filter due to parallel processing by the N local
filters (in multiprocessor systems). An additional
speed increase is possible by using the local filters
as prefilters to “compress” the local sensor data
and thus reduce the master filter processing rate.
This data compression feature of the new federated
design is possible because the local filters perform
real, recursive filtering. They maintain their ability
to smooth the noise in a sequence of measurements
(though this ability is somewhat reduced by the process
noise multiplier 7;).
A major additional benefit of this data compression
technique is that the master filter is no longer locked
to the local filter processing cycles, but can operate
at a selectable multistep rate. Hence, the master
filter can readily accommodate differing local filter
measurement rates, and even asynchronous local
522
operations. It requires only that the local filter
solutions be time-tagged or otherwise projectable to
a common time point.
During these longer intervals between master filter
updates, the master filter state and covariance can be
propagated via (36) and (37), to provide higher rate
federated solution outputs for use by other real-time
system functions.
As discussed earlier, the master filter does ignore
some potentially usable system information (knowledge
of common process noises) when it combines local
solutions on a multistep basis. However, the resultant
estimates are quite valid, even though they fall short
of globally optimal accuracy. An attractive aspect of
this multistep implementation is that it permits a direct
trade between estimation accuracy and computation
load. In fact, the federated filter can readily be
implemented with a n adjustable cycle rate, so that its
performance can be tuned to different criteria under
different operating conditions (e.g., mission phases).
Fault Tolerance: The new federated filter
architecture supports system fault detection,
identification, and recovery at several levels. First,
since the local filters provide real, recursive filtering
capability, they can perform legitimate and effective
screening of local sensor measurements via residual
checks. (The process noise multiplier somewhat
reduces the “tightness” of these checks.) Second, the
master filter incorporates each local filter output
except number 1 as a measurement, and computes a
residual (master minus local state estimate) that can
likewise be used for fault detection. In (41), C ’ A x
is a vector of n; independent, unit-variance random
numbers, for which fault thresholds can readily be
defined.
Note also that the propagated master filter solution
(36), (37) can be employed to fault-check the number 1
filter solution used to start the fusion process in (38).
From a fault-tolerance viewpoint, the globally
optimal federated filter implementation does exhibit
one serious drawback. In particular, feeding back
the fused state estimates and covariances from the
master to the local filters introduces the possibility of
cross-contamination. A fault in one sensor subsystem,
if undetected by both the local and master filters, will
contaminate the fused solution. Feeding this solution
back to the local filters will then contaminate each of
them.
The new federated filter design provides a simple
but suboptimal solution to the cross-contamination
problem. This solution is simply to bypass the feedback
of fused state and covariance data from the master
to the local filters. In this implementation, the master
filter still optimally combines the local filter solutions
via the normal method. However, it does not then
transmit fused solution data back to the local filters.
While suboptimal, this approach is theoretically
correct, and all filter components maintain “honest”
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 26, NO. 3 MAY 1990
covariances. The resultant estimation accuracy falls
short of the global optimum, but still exceeds that of
any of the stand-alone local filters (provided the yi
factors are properly chosen). For many fault-tolerant
system applications, this level of accuracy is quite good
enough, and is more than offset by the increased fault
detection, isolation, and recovery capabilities.
VI.
CURRENT A N D FUTURE APPLICATIONS
The new federated filter design is applicable to
both Type A and Type B systems (defined in Section
II), in one or more of its alternate implementations.
Type B systems represent idealized future applications
with total design flexibility, hence, the local filters
can be made to cooperate in support of global
performance criteria. For example, they can be
designed to accept feedback of fused state and
covariance data from the master filter, for maximum
estimation accuracy.
Type A systems represent nearer-term applications.
Here, the local filters comprise essentially fixed
designs, developed elseNhere for stand-alone
operation. These filters may provide some flexibility
in terms of adjustable model parameters (e.g., process
noise strengths or system time constants), but will not
accept data feedback from a master fusion filter.
The last federated filter implementation of the
previous section is quite suitable for this Type A
system application. While it cannot provide globaliy
optimal estimation accuracy, it does improve on
individual local filter performance. Furthermore, it
provides a very high degree of fault tolerance.
One may ask how these fixed local filters will be
affected by the yi x N multiplier on common process
noise components. The answer is that this multiplier
will generally be unnecessary. Most real-time filters
are designed with conservative (excess) process noise,
in order to keep the filter "open" to new measurement
information. Hence a process noise multiplier is
already present in the ratio of modeled to real noise
levels.
VII.
CONCLUSION
The new federated filter architecture developed in
this paper provides a number of significant advantages
for real-time distributed system applications:
1) flexible implementation to support a variety of
system requirements and facilitate performance
trades;
2) globally optimal estimation accuracy in single-step
implementation with feedback to local filters;
3) conservatively suboptimal estimation accuracy in
multistep implementation with feedback;
4) increased throughput due both to parallel local
filter processhg, and to sensor data compression
via effective local prefiltering;
5) multilevel fault detection, isolation and recovery
capability, highly fault tolerant in no-feedback
implementation with totally independent local filter
solutions;
6) minimal data transfer requirements: filter states and
covariances only;
7) efficient, numerically stable implementation in
covariance (or information) square root form;
8) simple real-time implementation due to
independent local filter operations, particularly in
no-feedback, fault-tolerant mode;
9) application to distributed systems with both fixed
(current) and cooperative (future) local filters.
Overall, this new federated filtering method seems
quite attractive for a variety of real-time system
applications, including distributed navigation systems.
Several of its alternate implementations warrant
further analysis and testing.
APPENDIX.
THEORETICAL FOUNDATION
This Appendix presents a more rigorous derivation
of the new federated filtering method outlined in
Section 111. We prove single-step global optimality
and define the multistep information loss. Consider the
sequential system dynamics and discrete measurement
processes described by (1) to (8). The globally optimal
state estimate x for this system minimizes the following
quadratic cost index [6], where S, W , and V are
square roots of the state error, process noise, and
measurement noise covariance matrices:
(All
There are N independent sensors designated by
index i. Index j = 1, k refers to successive time steps.
Rewrite (Al) so that all terms have index i partitions:
The matrices $0 and W;, represent i = l , N scalar
multiples of the original matrices SOand W,,
constructed so that their root-sum-squares equal the
CARLSON: FEDERATED SQUARE ROOT FILTER FOR DECENTRALIZED PARALLEL PROCESSES
5 23
transformation:
original values:
N
N
i=l
k
iel
J
Note that the multipliers 7i are 1.0 or larger;
their average inverse equals 1/N. The optimal
solution xi for each local problem can be determined
independently at each step k, by applying a series of
orthogonal transformations. The time propagation
transformation is performed first, and then the
measurement transformation. The result for local
partition i after the fist complete step, i.e., at k = 1+,
can be expressed as follows:
+ I %-'[U-
-x)]11 k .
(A4)
The measurement update values Si',Ci and 2: are
given by (34) and (39, and AZi = Zi - H& is the
calculated residual. The remaining matrices Si, R, and
E at k = 1 are determined as follows:
I
From (A4) we see that the locally optimal value of
xi is g,?, which zeroes the fist cost index term. The
second term is the irreducible measurement residual
(a constant). The third term is the time propagation
residual which, given x, can be zeroed by proper choice
of U.
The next step is the crucial one. To obtain the
globally optimal solution, we must find the single value
of x that minimizes the total cost index. This total is
the sum over i of the local terms (A4), which can be
written in similar form with composite N-row terms as
follows:
For simplicity, we omit the measurement residual
terms, which do not affect the result. The value
of x that minimizes the fist term in (A6) can
be determined by the following orthogonal
-c
(AS)
-x)II: + llrmII:
Choosing x to zero the first term of (AS) is the
apparent next step. However, we must still take into
account the second term in (A6). Because of the
initial construction (M),the N rows of this matrix are
all identical except for the scalar multipliers l/7;'*.
They thus readily reduce (RSS) to a single row, with
multiplier equal to [ ~ ~ ( 1 / 7 ~ / ~ ) 2=11.0:
'/*
,
(A6)1 = [ls:-l(k:
(Aq2
2
@;'(ki
These values of S,,,and k,,, are equivalent to the
optimal values in (15) and (16). Following this
transformation, the first term in (A6) becomes
= pv-1["
- P s - ' ( 2 - x)]11;5.
(A9)
This second term from (A6) thus reduces to exactly
the same form that would have resulted, had the single,
globally optimal estimate been propagated to begin
with. Thus, dividing the global problem into N local
problems does yield the globally optimal solution,
provided that 1) the common covariances are rescaled
per (M),and 2) the local solutions are recombined
after each measurement update cycle.
What happens if we don't recombine the local
estimates after each measurement cycle, but propagate
them as independent local estimates over several
cycles? The answer is evident from the second term
in (A6). After several local filter cycles, the rows
of that second term are no longer simply related
by scalar multipliers, but differ due to disjoint
measurement information from their local sensors.
Via a triangularization process similar to (A7), we
can reduce that N-row matrix to three rows, with this
general result:
(A6)2 = JIAiu+ Bix - cilli + IIB2x - C2)l2 + Ilcsll:.
(A101
Here, the first term can be zeroed by proper choice
of U, given x. The third is an irreducible residual. The
second term is of the same form as the measurement
terms H x - 2 in (Al). This "measurement" involves
x, and in theory could be combined with (A8) to
improve the estimate of x . A similar term arises at
each propagation step. Thus, operating the local filters
independently over several steps is equivalent to
ignoring an available measurement (of process noise
dimension p) at each such step. The resulting solution
is quite valid but conservatively suboptimal, as it is for
any filter that selectively uses part but not all of the
information potentially available to it.
REFERENCES
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Computation and transmission requirements for a
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IEEE Tramactwns on Automatic Control, AC-24, 2 (Apr.
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control’.
IEEE Transactwm on Automatic Control, AC-25, 3 (June
1980).
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and Verghese, G. C. (1982)
Combining and updating of local estimates and regional
maps along sets of one-dimensional tracks.
IEEE Tramactwns on Automatic Control, AC-21, 4 (Aug.
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Levy,B. U., et al. (1983)
A scattering framework for decentralized estimation
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Bierman, G. J., and Belzer, M. (1985)
A decentralized square root information filterhmoother.
In Proceedngs of the 24th IEEE Conference on Decision
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Kerr, T. H. (1985)
Decentralized filtering and redundancy
managementKailure detection for multisensor integrated
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Neal A. Carlson earned his B.S.E. in aeronautical engineering from Princeton
University, Princeton, NJ and his Ph.D. in aeronautics and astronautics from
Massachusetts Institute of Technology, Cambridge, MA.
Dr. Carlson is the founder and president of Integrity Systems, a small
aerospace engineering firm located in Winchester, MA. Integrity Systems
specializes in highly reliable avionics systems, software testing tools, and real-time
software. Neal worked for Intermetrics in Cambridge, MA from 1970 to 1983,
where he focused on integrated navigation systems, fault-tolerant avionics, and
Kalman filtering. H e was a major contributor to the Space Shuttle fault-tolerant
avionics system architecture, and to the GPS Phase I user navigation system.
His triangular square root filter formulation was the first stable yet efficient
mechanization of the Kalman filter, and the forerunner of today’s widely used
U-D mechanization. With Integrity Systems since 1983, Neal has developed a new
distributed Kalman filtering method for multisensor navigation systems. This new
method provides notable advantages over conventional methods in speed, fault
tolerance, and real-time system simplicity.
CARLSON: FEDERATED SQUARE ROOT FILTER FOR DECENTRALIZED PARALLEL PROCESSES
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